Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T19:34:26.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal Functions: Lp Estimates

Published online by Cambridge University Press:  20 November 2018

Huaihui Chen  and
Paul M. Gauthier
Huaihui Chen
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210024, P.R. China
Paul M. Gauthier
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

Keywords

30D4530F35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 1 , 01 February 1997 , pp. 55 - 73
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Ahlfors, L.V., Conformal Invariants, McGraw-Hill, New York, 1973.Google Scholar
2. Aulaskari, R., Hayman, W.K., and Lappan, P., An integral criterion for automorphic and rotation automorphic functions, Ann. Acad. Sci. Fenn., Series A. I. 15(1990), 201–224.Google Scholar
3. Aulaskari, R. and Lappan, P., An integral condition for harmonic normal functions, Complex Variable. 23(1993), 213–219.Google Scholar
4. Chen, H. and Gauthier, P.M., On strongly normal functions, Canad.Math. Bull., to appear.Google Scholar
5. Dufresnoy, J., Sur l’aire sphérique décrite par les valeurs d’une fonction méromorphe, Bull. Sci. Math. 65(1941), 214–219.Google Scholar
6. Hayman, W.K., Some applications of the transfinite diameter to the theory of functions, Journal d’Analyse Mathématiqu. 1(1951), 155–179.Google Scholar
7. Lappan, P., Some sequential properties of normal and non-normal functions with applications to automorphic functions, Comm. Math. Univ. Sancti Paul. 12(1964), 41–57.Google Scholar
8. Lappan, P., Some results on harmonic normal functions, Math. Z. 90(1965), 155–159.Google Scholar
9. Lehto, O. and Virtanen, K.I., Boundary behaviour and normal meromorphic functions, ActaMath. 97(1957), 47–65.Google Scholar
10. Lohwater, A.J. and Ch. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn., Series A. I. Math. 550(1973).Google Scholar
11. Pommerenke, Ch., Estimates for normal meromorphic functions, Ann. Acad. Sci. Fenn., A. I. Math. 476(1970).Google Scholar
12. Pommerenke, Ch. , On normal and automorphic functions, Michigan Math. J. 21(1974), 193–202.Google Scholar